Calibrated geometry


The mathematical field of differential geometry, a calibrated manifold is a Riemannian manifold of dimension n equipped with a differential p-form φ which is a calibration, meaning that:
Set Gx =. Let G be the union of Gx for x in M.
The theory of calibrations is due to R. Harvey and B. Lawson and others. Much earlier Edmond Bonan introduced G2-manifold and Spin-manifold, constructed all the parallel forms and showed that those manifolds were Ricci-flat. Quaternion-Kähler manifold were simultaneously studied in 1967 by Edmond Bonan and Vivian Yoh Kraines and they constructed the parallel 4-form.

Calibrated submanifolds

A p-dimensional submanifold Σ of M is said to be a calibrated submanifold with respect to φ if TΣ lies in G.
A famous one line argument shows that calibrated p-submanifolds minimize volume within their homology class. Indeed, suppose that Σ is calibrated, and Σ ′ is a p submanifold in the same homology class. Then
where the first equality holds because Σ is calibrated, the second equality is Stokes' theorem, and the inequality holds because φ is a calibration.

Examples